Problem: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{-6z^2 + 48z - 42}{z^3 - 17z^2 + 70z}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {-6(z^2 - 8z + 7)} {z(z^2 - 17z + 70)} $ $ a = -\dfrac{6}{z} \cdot \dfrac{z^2 - 8z + 7}{z^2 - 17z + 70} $ Next factor the numerator and denominator. $ a = - \dfrac{6}{z} \cdot \dfrac{(z - 7)(z - 1)}{(z - 7)(z - 10)}$ Assuming $z \neq 7$ , we can cancel the $z - 7$ $ a = - \dfrac{6}{z} \cdot \dfrac{z - 1}{z - 10}$ Therefore: $ a = \dfrac{ -6(z - 1)}{ z(z - 10)}$, $z \neq 7$